Can children get sick by living with someone who is getting radiation therapy?

No. With regular radiation given by machines a patient does not come home with any radiation and everyone is perfectly safe. In certain cases a patient is given radioactive substances and there are risks to children and pregnant women. For these patients there will be time and distance rules given in a written format to the patient and the family.
No. With external beam radiation therapy the patient does not have any radioactivity when they leave the clinic so they can not expose others to radiation. There are some considerations to keep in mind with low dose rate "seed" prostate brachytherapy.